The model parameters and standard errors are exactly the same, but
the model fit is better under -oglm-. The

Not really. The hypotheses being tested are different. There are 2
equations in the model (typically called choice and variance, or else
location and scale). oglm is doing a likelihood ratio chi-square
test of whether the coefficients in both equations all equal
zero. complogit is only testing the coefficients in the first
equation and is using a Wald test. (Note that the d.f. reported by
the two programs are different.) With oglm, it is easy enough to do
other Wald or LR tests if you don't happen to like the one that is reported.

key issue, however, is the value of the $\{delta}$ parameter.
under -gplogit-, $\{delta} < 0$, implying that residual variation
is larger amongst blacks than amongst non-blacks. Under -oglm-,
$\{delta} > 0$, which implies exactly the opposite. This throws up
two follow-up questions:

Not so. As noted in one of my other followup messages, a little bit
of algebra can switch you back and forth between Allison's delta and
oglm's lnsigma, but they are not exactly the same thing. Further,
whatever tests you look at (z values, Wald or LR chi-square tests)
you conclude that the residual variances do not significantly differ by race.

(1) If both of these achieved significance here, how on Earth do you
decide whether or not to include an interactive term in the model?

None of the analyses presented so far have said anything about
interaction terms; they've only addressed whether residual variation
differs across groups, and the answer seems to be no. If you now
want to test interaction terms involving race, go ahead; and plain
old -logit- will probably be adequate for your needs.

I'm not sure if this clearly answers your questions or not! If not,
all sorts of reading materials are available at